Periodic and solitary wave solutions to the Fornberg-Whitham equation

New travelling wave solutions to the Fornberg-Whitham equation are investigated. They are characterized by two parameters. The expresssions for the periodic and solitary wave solutions are obtained.


Introduction
Recently, Ivanov [1] investigated the integrability of a class of nonlinear dispersive wave equations where and α, β, γ, κ, ν are real constants.
The important cases of Eq.(1.1) are: The hyperelastic-rod wave equation has been recently studied as a model, describing nonlinear dispersive waves in cylindrical compressible hyperelastic rods [2]- [7]. The physical parameters of various compressible materials put γ in the range from -29.4760 to 3.4174 [2,4].
The Fornberg-Whitham equation appeared in the study qualitative behaviors of wave-breaking [28]. It admits a wave of greatest height, as a peaked limiting form of the travelling wave solution [29], u( where A is an arbitrary constant. It is not completely integrable [1]. The regularized long-wave or BBM equation (1.6) and the modified BBM equation have also been investigated by many authors [30]- [38].
Many efforts have been devoted to study Eq.(1.2)-(1.4),(1.6) and (1.7), however, little attention was paid to study Eq.(1.5). In [39], we constructed two types of bounded travelling wave solutions u(ξ)(ξ = x − ct) to Eq.(1.5), which are defined on semifinal bounded domains and called kink-like and antikinklike wave solutions. In this paper, we continue to study the travelling wave solutions to Eq.(1.5). Following Vakhnenko and Parkes's strategy in [19], we obtain some periodic and solitary wave solutions u(ξ) to Eq.(1.5) which are defined on (−∞, +∞). The travelling wave solutions obtained in this paper are obviously different from those obtained in our previous work [39]. To the best of our knowledge, these solutions are new for Eq.(1.5). Our work may help people to know deeply the described physical process and possible applications of the Fornberg-Whitham equation.
The remainder of the paper is organized as follows. In Section 2, for completeness and readability, we repeat Appendix A in [19], which discuss the solutions to a first-order ordinary differential equaion. In Section 3, we show that, for travelling wave solutions, Eq.(1.5) may be reduced to a first-order ordinary differential equation involving two arbitrary integration constants a and b. We show that there are four distinct periodic solutions corresponding to four different ranges of values of a and restricted ranges of values of b. A short conclusion is given in Section 4.

Solutions to a first-order ordinary differential equaion
This section is due to Vakhnenko and Parkes (see Appendix A in [19]). For completeness and readability, we repeat it in the following.
Consider solutions to the following ordinary differential equation and ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 are chosen to be real constants with Following [40] we introduce ζ defined by Eq.(2.4) has two possible forms of solution. The first form is found using result 254.00 in [41]. Its parametric form is with w as the parameter, where In (2.5) sn(w|m) is a Jacobian elliptic function, where the notation is as used in Chapter 16 of [42]. Π(n; w|m) is the elliptic integral of the third kind and the notation is as used in Section 17.2.15 of [42].
The solution to (2.1) is given in parametric form by (2.5) with w as the parameter. With respect to w, ϕ in (2.5) is periodic with period 2K(m), where K(m) is the complete elliptic integral of the first kind. It follows from (2.5) that the wavelength λ of the solution to (2.1) is where Π(n|m) is the complete elliptic integral of the third kind.
The second form of solution of (2.5) is found using result 255.00 in [41]. Its parametric form is where m, p, w are as in (2.6), and The solution to (2.1) is given in parametric form by (2.10) with w as the parameter. The wavelength λ of the solution to (2.1) is Eq.(1.5) can also be written in the form where a and b are two arbitrary integration constants.
In the following, suppose that a < 2 and a = 0 such that f (ϕ) has three distinct stationary points: ϕ L , ϕ R , 0 and comprise two minimums separated by a maximum. Under this assumption, Eq. (1) a < 0 In this case ϕ L < 0 < ϕ R and f (ϕ L ) < f (ϕ R ). For each value a < 0 and 0 < b < b R (a corresponding curve of f (ϕ) is shown in Fig.1(a)), there are periodic inverted loop-like solutions to Eq.(3.3) given by (2.5) so that 0 < m < 1, and with wavelength given by (2.8); see Fig.2(a) for an example.
The case 16 9 < a < 2 and b = b L (a corresponding curve of f (ϕ) is shown in Fig.1(h)) corresponds to the limit ϕ 1 = ϕ 2 = ϕ L so that m = 1, and then the solution is a hump-like solitary wave given by (2.13) with ϕ L < ϕ ≤ ϕ 3 and On the above, we have obtained expressions of parametric form for periodic and solitary wave solutions ϕ(ξ) to Eq.(3.3). So in terms of u = ϕ(ξ)+c, we can get expressions for the periodic and solitary wave solutions u(ξ) to Eq.(1.5).

Conclusion
In this paper, we have found expressions for new travelling wave solutions to the Fornberg-Whitham equation. These solutions depend, in effect, on two parameters a and m. For m = 1, there are inverted loop-like (a < 0), single peaked (a = 16 9 ) and hump-like ( 16 9 < a < 2) solitary wave solutions. For m = 1, 0 < a < 16 9 or 0 < m < 1, a < 2 and a = 0, there are periodic hump-like wave solutions.